A Matrix Approach to Hypergraph Stable Set and Coloring Problems with Its Application to Storing Problem

Abstract

This paper considers the stable set and coloring problems of hypergraphs and presents several new results and algorithms using the semitensor product of matrices. By the definitions of an incidence matrix of a hypergraph and characteristic logical vector of a vertex subset, an equivalent algebraic condition is established for hypergraph stable sets, as well as a new algorithm, which can be used to search all the stable sets of any hypergraph. Then, the vertex coloring problem is investigated, and a necessary and sufficient condition in the form of algebraic inequalities is derived. Furthermore, with an algorithm, all the coloring schemes and minimum coloring partitions with the given q colors can be calculated for any hypergraph. Finally, one illustrative example and its application to storing problem are provided to show the effectiveness and applicability of the theoretical results.